10.3: Important Examples

Definition: identity function (on a set \(A\))

the function \(A \to A\) defined by \(a \mapsto a\)

Definition: \(id_A: A\rightarrow A\)

the identity function on on set \(A\)

Note \(\PageIndex{1}\)

An identity function is always a bijection.

Definition: inclusion function (on subset \(A \subseteq X\))

the function \(A \to X\) defined by \(a \mapsto a\)

Check your understanding.

Do you understand the difference between the definitions of identity function and inclusion function?

Definition: \(\iota_A^X : A \rightarrow X\)

the inclusion function on subset \(A \subseteq X\)

Note \(\PageIndex{2}\)

An inclusion function is always an injection.

Definition: Projection Functions (on a Cartesian product \(A \times B\))

the functions \(A \times B \to A\) and \(A \times B \to B\) defined by \((a,b) \mapsto a\) and \((a,b) \mapsto b\)

Definition: \(\rho_A : A \times B \rightarrow A\)

the projection function onto the first factor \(A\) in the Cartesian product \(A \times B\)

Definition: \(\rho_B : A \times B \rightarrow B\)

the projection function onto the second factor \(B\) in the Cartesian product \(A \times B\)

Example \(\PageIndex{1}\)

Consider \((\dfrac{1}{2},\pi) \in \mathbb{Q} \times \mathbb{R}\text{.}\) Then

\begin{align*} p_{\mathbb{Q}} \left( \dfrac{1}{2}, \pi \right) & = \dfrac{1}{2}, & p_{\mathbb{R}} \left( \dfrac{1}{2}, \pi \right) & = \pi \text{.} \end{align*}

Extend.

We may of course similarly define a projection function on a Cartesian product with any number of factors. Write

\begin{equation*} \rho_i: A_1 \times A_2 \times \cdots \times A_n \rightarrow A_i \end{equation*}

to mean the projection function onto the \(i^{th}\) factor \(A_i\) in the Cartesian product

\begin{equation*} A_1 \times A_2 \times \cdots \times A_n\text{.} \end{equation*}

Alternatively, we may write

\begin{equation*} \text{proj}_i: A_1 \times A_2 \times \cdots \times A_n \rightarrow A_i \end{equation*}

for this function.

Note \(\PageIndex{3}\)

A projection is always surjective (except possibly when one or more of the factors in the Cartesian product is the empty set).

Definition: Restricting the domain

the “induced” function \(A \to Y\) created from function \(f: X\rightarrow Y\) and subset \(A \subseteq X\) by “forgetting” about all elements of \(X\) that do not lie in \(A\)

Definition: \(f \vert _A\)

restriction of function \(f: X\rightarrow Y\) to subset \(A \subseteq X\)

Definition: \(f\vert A\)

alternative domain restriction notation

Definition: \(\text{res} _A^X f\)

alternative domain restriction notation

Figure \(\PageIndex{1}\): A Venn diagram of restricting the domain of a function.

Example \(\PageIndex{2}\): Domain restriction.

For \(f: \mathbb{Z} \rightarrow \mathbb{N}\text{,}\) \(f(m) = \vert m \vert \text{,}\) we have \(f \vert _{\mathbb{N}} = \id_{\mathbb{N}}\text{.}\)

Checkpoint \(\PageIndex{1}\):Properties of restrictions.

Consider function \(f: X\rightarrow Y\) and subset \(A \subseteq X\text{.}\)

1. If \(f\) is injective, is \(f \vert _A\) injective?
2. If \(f \vert _A\) is injective, must \(f\) be injective?
3. Answer the previous two questions replacing “injective” with “surjective”.

Remark \(\PageIndex{1}\)

The concept of restricting the domain makes our previously defined concept image of a function on a subset unnecessary: for function \(f: X\rightarrow Y\) and subset \(A \subseteq X\text{,}\) the image of \(f\) on \(A\) is the same as the image of the restriction \(f \vert _A\text{.}\)

Definition: Restricting the codomain

the “induced” function \(X \to B\) created from function \(f: X\rightarrow Y\) and subset \(B \subseteq Y\) by “forgetting” about all elements of \(Y\) that do not lie in \(B\text{,}\) where \(B\) must contain the image of \(f\)

Figure \(\PageIndex{2}\): A Venn diagram of restricting the codomain of a function.

 

Example \(\PageIndex{3}\): Codomain restriction

Consider \(f: \mathbb{R} \rightarrow \mathbb{R}\text{,}\) \(f(x) = x^2\text{.}\) It would be more precise to write \(f: \mathbb{R} \rightarrow \mathbb{R}_{\geq 0} \\text{,}\) since \(x^2 \ge 0\) for all \(x \in \mathbb{R}\text{.}\)

Note \(\PageIndex{4}\)

If we restrict the codomain all the way down to the image set \(f(X)\text{,}\) the resulting map \(f : X \rightarrow f(X) \) is always surjective. In particular, if \(f: X \xrightarrow Y\) is injective, then by restricting the codomain we can obtain a bijection \(f : X \rightarrow f(X) \text{.}\)

Definition: Extension of a function

relative to function \(f: A \rightarrow B\) and superset \(X \supseteq A\text{,}\) a function \(g: X \rightarrow B\) so that \(g(a) = f(a)\) for all \(a \in A\)

Figure \(\PageIndex{3}\): A Venn diagram of a function extension.

Note \(\PageIndex{5}\)

The condition defining the concept extension function can be more succinctly stated as requiring function \(g: X \rightarrow B\) with \(A \subseteq X\) satisfy \(g \vert _A = f\text{.}\)

Example \(\PageIndex{4}\): Floor function.

Write \(\text{flr}:\mathbb{R} \rightarrow \mathbb{Z}\) to mean the floor function: for real input \(x\text{,}\) the output \(\text{flr} (x)\) is defined to be the greatest integer that is less than or equal to \(x\text{.}\) Usually we write

\begin{equation*} \text{flr} (x) = \lfloor x \rfloor \text{.} \end{equation*}
As every integer is less than or equal to itself, we have \(\text{flr} (z) = z\) for every \(z \in \mathbb{Z}\text{.}\) This says that the floor function is an extension of the identity function \(\id_{\mathbb{Z}}\text{.}\)

One of the most common ways to extend a function to a larger domain is to pick an appropriate constant value in the codomain to assign to all “new” inputs in the enlarged domain.

Definition: Extenstion by zero

relative to function \(f: A \rightarrow Z\) and superset \(X \supseteq A\text{,}\) where \(Z\) is a set of “numbers” containing a zero element, the extension function \(g: X \rightarrow Z\) defined by

\begin{equation*} g(x) = \begin{cases} f(x)\text{,} & x \in X \text{,} \\ 0\text{,} & \text{otherwise.} \end{cases} \end{equation*}

Example \(\PageIndex{5}\): Extending the identity function by zero.

Define \(\widetilde{\id}_{\mathbb{Z}} : \mathbb{R} \rightarrow \mathbb{Z}\) by

\begin{equation*} {\widetilde{\id}}_{\mathbb{Z}}(x) = \begin{cases} x\text{,} & x \in \mathbb{Z} \text{,} \\ 0\text{,} & \text{otherwise.} \end{cases} \end{equation*}
Then \(\widetilde{\id}_{\mathbb{Z}}\) is the extension by zero of the identity function \(\id_{\mathbb{Z}}\text{.}\)

Compare.

Example \(\PageIndex{4}\) also involved an extension of the identity function \(\id_{\mathbb{Z}}\) — was it an extension by zero?